Theorem dedth3v 2385

Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 2384.

Hypotheses

Ref	Expression
dedth3v.1	|- (A = if(ph, A, D) -> (ps <-> ch))
dedth3v.2	|- (B = if(ph, B, R) -> (ch <-> th))
dedth3v.3	|- (C = if(ph, C, S) -> (th <-> ta))
dedth3v.4	|- ta

Assertion

Ref	Expression
dedth3v	|- (ph -> ps)

Proof of Theorem dedth3v

Step	Hyp	Ref	Expression
1		dedth3v.4	. 2 |- ta
2		iftrue 2366	. . . . 5 |- (ph -> if(ph, A, D) = A)
3	2	eqcomd 1480	. . . 4 |- (ph -> A = if(ph, A, D))
4		dedth3v.1	. . . 4 |- (A = if(ph, A, D) -> (ps <-> ch))
5	3, 4	syl 10	. . 3 |- (ph -> (ps <-> ch))
6		iftrue 2366	. . . . 5 |- (ph -> if(ph, B, R) = B)
7	6	eqcomd 1480	. . . 4 |- (ph -> B = if(ph, B, R))
8		dedth3v.2	. . . 4 |- (B = if(ph, B, R) -> (ch <-> th))
9	7, 8	syl 10	. . 3 |- (ph -> (ch <-> th))
10		iftrue 2366	. . . . 5 |- (ph -> if(ph, C, S) = C)
11	10	eqcomd 1480	. . . 4 |- (ph -> C = if(ph, C, S))
12		dedth3v.3	. . . 4 |- (C = if(ph, C, S) -> (th <-> ta))
13	11, 12	syl 10	. . 3 |- (ph -> (th <-> ta))
14	5, 9, 13	3bitrd 544	. 2 |- (ph -> (ps <-> ta))
15	1, 14	mpbiri 194	1 |- (ph -> ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956  ifcif 2361

This theorem is referenced by:  projlem7 9192

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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